Cantor Confusion
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From: mueckenh on 9 Oct 2006 10:20 Virgil schrieb: > In article <1160295281.279569.143920(a)m7g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > If a set of axioms yields the theorems A and nonA, then this set is > > useless. The axioms of ZFC yield the theorems "the vase is empty at > > noon" and "the vase is not empty at noon". > > Not so. One has to add "Mueckenh"'s, or at least other assumptions, to > ZFC to get a nonempty vase at noon. > > The only ZFC analysis coincides with: > > Let A_n(t) be equal to > 0 at all times, t, when the nth ball is out of the vase, > 1 at all times, t, when the nth ball is in the vase, and > undefined at all times, t, when the nth ball is in transition. > > Note that noon is not a time of transition for any ball, though it is a > cluster point of such times. > > let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase > at any non-transition time t. > > B(t) is clearly defined and finite at every non-transition point, as > being, essentially, a finite sum at every such non-transition point. > > Further, A_n(noon) = 0 for every n, so B(noon) = 0. > Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 > > There is no ZFC-compatible argument for having a non-empty vase at or > after noon that does not require assumptions beyond those of ZFC. > > And it is with those additional assumptions that "Mueckenh" and others > make, that the conclusions of ZFC conflict. The assumption is that "lim{n->oo} 9n = 0" is wrong. If this assumption is not wrong in ZFC then ZFC is useless. Regards, WM
From: mueckenh on 9 Oct 2006 10:30 Virgil schrieb: > In article <1160308871.194701.44520(a)c28g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > 0) There is a bijection between the set of balls entering the vase and > > |N. > > When? Independent of time, all the balls are enumerated. > > > 1) There is a bijection between the set of escaped balls and |N. > > When? Independent of time, the balls are enumerated. But the time t can be enumerated by the balls leaving the vase. > > > 2) There is a bijection between (the cardinal numbers of the sets of > > balls remaining in the vase after an escape)/9 and |N. > > This does not occur ever. The contents of the vase is A(t) = 9t. It will no be reset for t --> oo. Lim {t-->oo} 9t = 0 is as false as Lim {k-->oo} Sum{1 to k} = 0. Regards, WM
From: mueckenh on 9 Oct 2006 10:37 Dik T. Winter schrieb: > The balls in vase problem suffers because the problem is not well-defined. > Most people in the discussion assume some implicit definitions, well that > does not work as other people assume other definitions. How do you > *define* the number of balls at noon? You can not use limits, because the > limit does not exist when you use standard mathematics. But we can safely say that lim{n-->oo}n = 0 is false. lim{n-->oo}n can be estimated by lim{n-->oo} 1/n = 0. So using standard > definitions there is no answer. More precise, given the sequence of sets: > {1, ..., 10) > {2, ..., 20} > {3, ..., 30} > etc., is there a limit? Well, no, there is no defined limit unless you > define what a limit of sets looks like. I have never seen a definition > that tells me how the limit of a sequence is defined. The limit of the > size of the sets also gives no answer, because that limit does not exist. > Strange enough, when somebody goes on to define things, *you* question his > definitions, rather than the result. The limit {1,...,n} for n-->oo is N, if N does exist. Regards, WM
From: mueckenh on 9 Oct 2006 10:42 Dik T. Winter schrieb: > In article <1160308871.194701.44520(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > You got it several times already. According to the axiom of infinity > > > > the set of all natural numbers does exist. And with it the following > > > > statements are true: > > > > > > > > 1) Before noon every ball comes out of the vase. At noon the vase is > > > > empty. > > > > 2) Before and at noon there are more balls in the vase than have come > > > > out. > > > > > > How do you translate the words of the problem into mathematics? > > > > 0) There is a bijection between the set of balls entering the vase and > > |N. > > 1) There is a bijection between the set of escaped balls and |N. > > 2) There is a bijection between (the cardinal numbers of the sets of > > balls remaining in the vase after an escape)/9 and |N. > > How do you *define* division between cardinal numbers? "/9" in (2) is completely negligible. Therefore no division need be defined. But, in principle, division by a finite cardinal number is defined. Also it is easy to estimate: X/9 = X*(1/9) =< X. More is not required here. Regards, WM
From: Jesse F. Hughes on 9 Oct 2006 10:46
"William Hughes" <wpihughes(a)hotmail.com> writes: > What? You think that something can sound unbelievable but still > be true? > > If something sounds unbelievable, how do we tell if it is true or > not? We ask Han. Duh. -- Jesse F. Hughes "When my brain begins to reel from my literary labors, I make an occasional cheese dip." -- Ignatius J. Reilly |